Characterization of unsteady double-diffusive mixed convection flow with soret and dufour effects in a square enclosure with top moving lid

1. 1.      Introduction

The convection simultaneously driven by temperature and concentration gradients are often called either double-diffusive or thermo-solutal convection. Double-diffusive convection is an attractive subject due to its wide scientific applications such as oceanography, astrophysics, geology, biology and chemical processes [1]. This aspect of fluid dynamics has been gained considerable attentions from the both of theoretical and experimental researchers because of its importance and wide practical applications such as electronic device cooling, multi-shield structures used for nuclear reactors, float gas production, crystal growth, drying processes, chemical reactors, and many others [2-6]. Based on that, wide advanced models as well as research methods have been developed to better understand the physical phenomena in involved in double-diffusive convection. Among the literature published on this subject, Lee & Hyun [8], and Hyun & Lee [9], numerically studied the double-diffusive convection in a rectangular enclosure with aiding and opposing temperature and concentration gradients. Their solution was significantly satisfied the experimental results. Oztop & Dagatekin numerically investigated the steady mixed convection in a two-sided lid-driven enclosure [10]. Results elucidate that the heat transfer enhances as Richardson number  value decreases. Al-Amiri et al., utilized numerical simulations to look through the steady double-diffusive convection in a square lid-driven cavity [11]. Results demonstrate that heat transfer enhances as the buoyancy ratio increases. Thermo-solutal convection with temperature and concentration gradients at the same time in a rectangular enclosure was studied by Qin et al, [12]. They used a high-order compact scheme in their study. Jena et al., researched on the transient process of buoyancy-opposed thermo-solutal convection of micropolar fluids [13]. Bhattacharya and Das also used numerical techniques to investigate the steady thermo-solutal natural convection flow inside a usual lid-driven cavity [14]. It has been shown that Rayleigh number is an important parameter in the heat transfer variation. Wang et al., performed regularized Lattice Boltzmann method (LBM) to study thermo-solutal convection in the vertical cavity [15]. It has been shown that heat and mass transfer are attenuated as the cavity aspect ratio enhances.

It is well known that the fluid flow is simultaneously affected by temperature and concentration gradients as well in the double-diffusive convection flow. In some cases, the extra thermal and mass diffusivity called Dufour and Soret effects, respectively, affect on the thermo-solutal flow characteristics to some extent. Soret effect is the extra mass diffusivity caused by the temperature gradient, while Dufour effect refers to the energy flux created by the concentration gradient. The Soret and Dufour effects are called SD-effects hereinafter for the sake of brevity. SD-effects are ignored in many cases due to their order of magnitude respect to the effects described by Fourier’s and Fick’s laws. Moreover, they have usually considered as the second-order phenomena. Nevertheless, in some engineering and industrial applications such as chemical reactors, solidification of binary alloys, groundwater pollutant migration, hydrology and geosciences, when temperature and concentration gradients are large enough, the SD-effects could not be ignored and should be taken into account to complete the accurate simulation. In those cases, the temperature and concentration equations become coupled with each other. Recently, some investigators have been conducted numerical and analytical studies to study double-diffusive convection when SD-effects were not neglected.

Malashetty & Gaikward studied numerically the influence of SD-effects on thermo-solutal convection in an unbounded vertically stratified system [16]. Rebai et al., investigate double-diffusive convection in a square cavity filled with binary fluid mixture using the both numerical and analytical methods [17]. Soret effect was just considered by them. Bhuvaneswari et al., performed numerical simulations to investigate mixed convection flow with just Soret effect in a regular two-sided lid-driven square enclosure [18]. They looked into the relation of the lid's movement direction and transport phenomena, and found that both of heat and mass transfer are attenuated if the walls move in the same directions. Actually, they did not consider the influence of Soret effect alone, because all of their numerical simulations performed at a constant Soret coefficient. Wang et al., used an unsteady numerical model to research on the influence of SD-effects on thermo-solutal convection in a horizontal rectangular enclosure [19]. Results show that heat and mass transfer increase as the aspect ratio decreases. Their simulations performed in a stable cavity with no moving lid(s) and so they did not study the effects of shear forces on the double-diffusive convection. Recently, Ren & Chen utilized the LBM to study double-diffusive convection in a vertical enclosure with SD-effects [20]. They found that the average Nusselt and Sherwood numbers were increased with increasing Rayleigh number, Prandtl number, Lewis number, Soret and Dufour coefficients. Kefayati also used the LBM to examine double-diffusive convection with SD-effects in an inclined porous cavity [21]. The results prove that heat and mass transfer are sensitive greatly to the inclination angle. Wang et al., utilized an accurate finite volume method based on SIMPLE algorithm to investigate numerically an oscillatory double-diffusive convection in a horizontal cavity with SD-effects [22]. They found that double-diffusive convection develops from steady state convection dominated to chaotic flow as buoyancy ratio increases.

In all of the above studies, a constant set of Soret and Dufour coefficients was assumed during the simulations. On the other hand, to the author's best knowledge, the effect of SD-effects on the various modes of heat transfer has not been analysed yet. To be more precise, the contribution of each modes of heat transfer and in particular, the influence of extra mass and heat diffusions on those contributions have not been studied yet. In addition, although the kinetic energy is a key factor in the design and optimization of thermal systems, the influence of SD-effects on the total energy of the thermo-solutal systems has not been considered, yet. Based on the above story, the main purpose of the present study is to characterize the unsteady thermo-solutal convection with SD-effects in one hand, and analyse the influence of SD-effects on the various modes of heat transfer as well as the total energy of the system in the other hand.

1. 2.      Model Description

The physical model configuration which consists of a square enclosure with just top moving wall in its own plane at constant velocity , is displayed in Fig. 1. The top and bottom walls are maintained isothermally at uniform temperatures  and   respectively, . The opposite boundary conditions are assumed for concentration at top and bottom walls, when those are maintained at concentrations  and  correspondingly, . The vertical walls are assumed adiabatic and impermeable. This illustration creates a gravitationally-unstable temperature and concentration gradients and results in a thermo-solutal/double-diffusive convection.

Fig.  1: Schematic diagram of the computational model.

The enclosure has an aspect ratio of unity () and dry air is assumed as the working fluid, . Fluid is also assumed Newtonian and incompressible except for the density in the buoyancy term of the momentum equation in the vertical direction, according to the Bousinesq approximation. Accordingly, the density variation due to both temperature and concentration gradients can be written as:

where  and,  are the thermal and concentration expansion coefficients, respectively. With these assumptions, the fundamental governing equations, including continuity, momentum, energy, and concentration (mass) equations can be expressed as:

where , , , ,  ,  are kinematic viscosity, gravity acceleration, thermal diffusivity, diffusion coefficient, Soret and Dufour coefficients, respectively. The governing equations are then non-dimensionalized using the following dimensionless variables:

where  is the characteristic time ,  is the characteristic pressure ,  is the characteristic temperature, and  is the characteristic particle concentration. Following Barletta  Zanchini [23], the characteristic temperature and concentration are assumed as  and , respectively. Therefore, the dimensionless form of governing equations are:

The buoyancy ratio , Lewis number , Richardson number , Dufour coefficient , and Soret coefficient , are defined as:

Richardson number value ususally illustrates the importance of natural convection relative to the forced convection, whereby the flow is dominated by forced and natural convction when  and , respectively. However, the boundary conditions in the dimensionless form are so:

To examine the heat and mass transfer within the enclosure, the average Nusslet and Sherwood numbers on the horizontal walls with maximum temperature and concentration are examined. For this purpose, the local Nusselt and Sherwood numbers along the horizontal bottom and top walls, respectively, are defined as:

                      (11)

                      (12)

The average Nusselt and Sherwood numbers then can be calculated as:

          (13)

 It should be noted that the different boundary conditions are implied for temperature and concentration to better understand the influence of the extra available diffusions on the fluid characteristics and transport phenomena. To better understand and discuss the heat transport processes, the different modes of heat transfer, i.e. conductive and convective modes,  across the enclosure are also examined by the relation proposed by Cheng [24]. For this purpose, Nusselt number along the vertical mid-plane of the enclosure is evaluated by the following equation:

                                       (14)

where the first and second terms of this equation represent the contributions of heat transfer because of the conductive and convective modes, respectively. Further examination of the total kinetic energy is also implemented in this study. It is calculated using the expression proposed before by Goyan through [25]:

                                   (15)

This equation is calculated on all of the grid nodes and averaged during final 1000 final time steps. The temporal variation of total kinetic energy is also analyzed in further to better examine the stationary state conditions.

1. 3.       Numerical Approach

The dimensionless governing equations, Eqs. (6)-(10), are firstly discretized on a staggered grid by a finite volume method developed by Patankar [26]. The convection terms are  discretized using the QUICK scheme, while a second-order Adams-Bashforth explicit scheme is implemented for the unsteady terms. The SIMPLE algorithm is then employed to solve the discretized equations. The effect of concentration is also taken into account by using pressure correction method to obtain the real velocity field. The averaged Sherwood and Nusselt numbers are calculated using Simpson's integration rule.

 The validation procedure of the utilized method had to be done in order to check the code credibility. This is imposed as well as the convergence and grid independency tests in further steps. The time step is set close to  during all simulations similar to the study of Ouertatani et al. [27]. The heat and mass transfer characteristics in addition to the fluid flow patterns have been reported when the steady state conditions are achieved. The unsetady patterns of the studied thermo-solutal system are illustrated in Fig. 2, where the temporal variations of isotherm and isoconcentration contours are presented. It can be seen that after , steady state conditions are achieved.  The steady state conditions are also examined by the investigation of typical temporal variations of total kinetic energy in Fig. 3. It is observed that after an initial unsteadiness, double-diffusive convection in various regimes become as steady as the total kinetic energy attains a canstant value. The convergence of the numericall results is also employed and the following criterion is satisfied on each time step.

Here, the generic variable  illustrates the set of , , , or , while  represents the iteration number in each individual time step. The subscript sequence  represents the space coordinates of the grid node. The simulations were performed for three various uniform grids, i.e. , , and , for an especific case when , and , and then the results were compared to gether in order to sure on grid independency. The obtained results have an excellent agreement and so,  grid was used in according to the proper accuracy as well as CPU consuming time.

considering simulated accuracy and CPU time in the range of variables adequate, results are obtained using node points .

The utilized method was validated against published results of Al-Amiri et al., [11] and Teamah & El-Maghlany [28] to sure on the accuracy of the future obtained results. Hence, numerical simulations in the absence of SD-effects, , were performed for double-diffusive mixed convection flow in a vertical square enclosure with the uniformly imposed high and low temperature as well as concentration along the lower and upper walls, respectively. Fig. 4 shows the streamlines, isotherms, and isoconcentration distributions with, , , , and , obtained by (a) present code, (b) Teamah & El-Maghlany [28], and (c) Al-Amiri et al. [11]. The figures show a good agreement between the results obtained by the present code and the others. Another test is conducted to check the accuracy of the utilized method, whereby the stream function values at the primary vortex location were computed for two different Reynolds number, i.e. 100 and 400, and are compared with the results obtained by Al-Amiri et al. [11] and Screibr & Keller [29] in Table 1. It can be seen that an excellent agreemnt was achieved between registered data results. Furtheremore, the average Nusselt number obtained by the present method and the results achieved by Sharif [30] and Malleswaran & Sivasankaran [31] are compared in Table 2. The results obtained by the present method have an acceptable aggreement with the available results, especially at the larger  and values.

Table 1. Comparison of primary vortex stream function.

1. 4.      Results and Discussion

As was mentioned before, the main aim of the present work is to characterize the thermo-solutal/double-diffusive convection flow, heat and mass transfer in a square enclosure and in the presence of Soret and Dufour effects. For this purpose, numerical simulations are carried out with the validated method for different Richardson numbers (), when  is kept constant at 10⁴ and Re is varied between 31.6 to 1000. Besides, Soret and Dufour effects are employed where  is kept constant at 0.25, and  is varied between 0 to 3. Schmidt number sets equal to Prandtl number, and so . The assumed value of Lewis number represents a same contribution for both of heat and mass transfer, whereby it makes an upportunity to stduy the influence of Richardson number individually. Buoyancy ratio , is also set to unity to consider similar effects for mass and thermal diffusions. Streamlines, isotherms, and isoconcentrations of the cases with different  and various sets of  coefficients are displayed in Fig. 5. As can be seen that an unicellular clockwise primary vortex almost occupies whole the cavity in all the studied cases.

Fig. 3. Temporal variations of total kinetic energies for various Richardson numbers when .

In fact, the fluid rises up from the heated bottom wall due to the thermal buoyancy forces and flows down along the cold side. The competition between forced flow introduced by top moving wall, solutal and thermal buoyancy forces have been formed a primary rotating cell. If the isotherm patterns of the cases with  are compared together, it can be seen that the concentration of thermal boundary layers near the heated wall reduces as Richardson number enhances. To help to better understand, the isotherm patterns of the cases with  are represented in Fig. 6. When Richardson number has its lowest value (Fig. 6(a)), Reynolds number is kept constant at 1000, the flow was dominated by forced convection introduced by top moving lid and the convection circulation was developed greatly. In the other words, shear forces push the convection to penetrate much deeper into the enclosure. A similar observation was reported before in the studies of Al-Amiri et al. [11], and Teamah & El-Maghlany [28], where the double diffusive mixed convection in the absence of SD-effects was investigated numerically.

However, with either decreasing Reynolds number or increasing Richardson number (Figs. 6(b)&(c)), the opposing action of thermal buoyancy forces against forced flow was pronounced, whereby the concentration of the thermal boundary layers in the vicinity of the hot wall was reduced. When  (Fig. 6(d)), the entire isotherm lines were became parallel to the horizontal walls. This configuration demonstrate that heat was transferred mostly by conduction mode, whereby the enclosure could be assumed as a quasi-conduction domain. It should be noted that the isotherm and isoconcentration patterns in the cases with  are almost similar due to the value of Lewis number, i.e. . This fact is not repeated when either the extra mass or thermal diffusion is introduced.

Table 2. Comparison of the average Nusselt numbers.

 Fig. 5 also depicts that at the lower and moderate values of Dufour coefficient, i.e.  0.25 and 1, the influence of SD-effects on fluid flow, isotherms, and isoconcentrations seems to be insignificant. In addition, the influence of SD-effects on even fluid flow can be ignored when Richardson number is large enough. Nevertheless, Figs. 5(c)&(d) show that the enhancement of Dufour coefficient could develop fluid flow and transport phenomena when flow is in the mixed or natural convection regime. The extra thermal diffusion within the enclosure, which was increased by increasing Dufour coefficient, is in the direction of concentration gradient, and so has an aiding and opposing action on the shear forces and thermal buoyancy forces, respectively. As it was mentioned earlier, the shear forces introduced by top moving wall, have a tendency to push the convection into the enclosure, whereby the thermal boundary layers are concentrated greatly near the hot wall with further decreasing of Richardson number. Therefore, the competition between forced convection on one side and thermal buoyancy forces as well as extra thermal diffusion introduced by SD-effects on the other side, forms thermal eddies near the heated wall. Those thermal eddies cause a distortion in the thermal boundary layers even if when the enclosure is a quasi-conductive domain. In addition, the extra thermal diffusion disturbs the equilibrium between mass and thermal diffusing, assumed before by implying, and so isotherms and isoconcentrations are not the same here.

Figs. 5(c)&(d) represents that when Richardson number was increased and so forced convection was ground by natural flow, the extra thermal diffusion along with the thermal buoyancy forces have formed the secondary eddies at the left bottom of the enclosure. It can be seen that the secondary eddy formation is augmented with increasing either Richardson number or Dufour coefficient. On the other hand, it seems that the variation of Dufour coefficient was also affected to some extent on the isoconcentration contours. This is attributed to the fact that the mass transfer rate was affected to some extent by convection activities.

Fig. 4. Streamlines, isotherms, and isoconcentrations for , , , and , obtained by (a) present method, (b) Teamah & El-Maghlany [28], and (c) Al-Amiri et al. [11].

Fig.  5: Streamlines (left column), isotherms (middle column), and isoconcentrations (right column) for (a) , (b) , (c) , and (d) , when  and .

Fig.  6: Isotherm patterns for (a) , (b) , (c) , and (d) , when  and .

Fig.  7: Isotherm patterns for (a) , (b) , (c) , and (d) , when  and .

Fig.  8: Isoconcentration patterns for (a) , (b) , (c) , and (d) , when  and .

Fig.  9: Horizontal mid-velocity profiles for (a) , (b) , (c) , and (d) . In all of cases .

Fig.  10: Vertical mid-velocity profiles for (a) , (b) , (c) , and (d) . In all of cases .

Fig. 11: Average (a) Nusselt, and (b) Sherwood numbers, when  for all cases.

Fig.  12: The vertical distributions of conduction mode of heat transfer along the mid-plane of the square enclosure for  (a) , (b) , (c) , and (d) , and the vertical distributions of convection mode of heat transfer along the mid-plane of the square enclosure for (a) , (b) , (c) , and (d) , where   in all of cases